Page 300 - Maxwell House

280 Chapter 6

adequate energy bellow some frequency called the cut-off frequency. As a result, the wave
“punishes such waveguide refusing” to carry active energy in it.
As soon as we start extending these results to any line, the difficulties begin. Evidently, there
is no problem to estimate the power P and stored electric or magnetic energy in any

area of arbitrary line. Simply put, we can measure or calculate the electric or magnetic field
distribution in line to evaluate this power and energy using Poynting’s theorem and define the
first three wave parameters in (6.8). The real problem is how to define uniquely the currents
and voltages in (6.7) or (6.8). There are no propagating currents or voltages in Maxwell’s
equations and Poynting’s theorem, the fields only. If so, the logical step is to express the electric
voltage and current through the fields (see (1.21) and (1.66) in Chapter 1) as

= ∫ ∘
� (6.9)
= ∮ ∘

Both values have a clear physical meaning being proportional to the work that can be produced
by electric or magnetic fields. The troubles are how to choose the integration contours in the
process of such work. It is trivial in the case of static and TEM mode electric or magnetic fields.
For example, we know that the numerical value of the first integral in (6.9) is determined by
the starting and ending points and independent on the path between them. The second one is
also unique being independent on the shape of contour encircling the current. As soon as the
EM field pattern is the same as static and does not depend on the frequency, we can use both
definitions in (6.9) and get the unique values. Therefore, the unique and exact equivalent
transmission line with particular impedance can be built for any line with TEM mode only and
quasi-TEM with some caution.

In all other cases, we need the engineering approach and reach the agreement on how to select
the path and contour in (6.9) that suits all users and applications. Evidently, such “wild”
freedom makes the unique calculation of characteristic impedance is nearly impossible. That is
why we put three different definitions of characteristic impedance in the last line of (6.8). All
three provide the same value for TEM-mode but entirely different for modes of more
complicated patterns. In general, the proper impedance definition primarily depends on the sort
of lumped elements connected to the line. At best, one may use the power and current
combination for series elements while for shunt ones the power and voltage definition is more
natural. In more complicated cases one may follow her/his engineering ingenuity and intuition.

6.1.3 Concept of Cutoff Wavelength / Frequency

Let us look a little dipper into the idea of cutoff wavelength we have mentioned before. Assume
that some free-of-loss line is completely shielded inside the perfectly conductive metal tube of
proper diameter and the wave in line carries both active and reactive power along the line.
According to the reactive energy balance (3.52), the propagating EM waves can carry the
reactive energy if and only if they are “nourished” by some inside ℑ� ()� or outside
ℑ� ()� reactive power sources. But the metal tube completely prevents the energy
Σ
penetration from outside, and we surely do not want to put any kind of extra sources inside.
Therefore, in any freely propagating wave = as we suggested getting the equations (6.7)

and (6.8). There is one exclusion. Suppose that EM wave bounces back and forth reflecting